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Although great care has been taken to provide accurate and current information, neither the

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for any loss, damage, or liability directly or indirectly caused or alleged to be caused bythis book. The material contained herein is not intended to provide specific advice or

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To Pili and Eladia, our wives


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Preface

Polymers differ from simple crystalline solids and simple liquids in that they have

length and molecular scales larger than atomic, a characteristic that gives themunusual properties. The underlying structure of a polymer is a long chain in which

one or more chemical motives repeat along the chain. Most of the skeletal bonds

of polymers are of a type that can rotate, giving rise to an unimaginably large

number of spatial conformations. As a result, statistical considerations must enter

into the description of even the simplest molecular chain. Moreover, the macro-

molecular nature of polymers vastly broadens the time scale for molecular adjust-

ments to external force fields. The macromolecular size of polymers makes them

suitable to develop materials that may combine great elasticity with great tough-

ness, fluidity with a solid-like structure, etc. As a result, polymers are ubiquitousin both nature and industry.

A relevant characteristic of polymers is their ability to withstand high

electric fields with negligible conduction due to the large energy differences

between the localized valence electronic states and the conduction band.

This characteristic coupled with favorable mechanical and processing properties

make polymers the obvious choice for insulating applications. However, the

versatility of polymers may expand their window of use to include the most

unexpected applications. Thus the coupling between the electric properties of

some polymers and their mechanical and thermal properties has led to importanttransducer applications. Moreover, in the last decades increased scientific and

technological attention has been given to the development of polymeric electric

conductors.


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This book is mainly concerned with the response of polymers to electric

fields. We have approached this subject in such a way that the book meets the

requirements of the beginner in the study of the electric properties of polymers aswell as those of experienced workers in other type of materials. The book is

divided into three parts: Part I deals with the physical fundamentals of dielectrics,

Part II with the relation between structure and equilibrium and dynamic dielectric

properties, and Part III with the electric response of special polymers to force

fields.

An understanding of the response of polymers to electric fields requires

knowledge of the basic physical properties of dielectrics, and how these prop-

erties are affected by molecular size is offered in Chapters 1 and 2. Chapter 1

deals with the interactions between dipoles and static electric fields and thedescription of theories that relate the static dielectric permittivity with the polar-

ity of low-molecular-weight amorphous compounds or monomers. These

theories are extended to high-molecular-weight chains and inter- and intra-

molecular dipolar correlations that depend on the molecular structure are

considered.

Most processes in nature are stochastic and their description in this

discussion requires the use of probability and averages. In Chapter 2, both the

Langevin and Schmulokowsky equations that describe the probability distribu-

tion in stochastic processes are introduced and further used in the description ofdielectric relaxations using the Debye and Onsager models. Attention is paid to

the relation between statistical mechanics and linear dielectric responses for

systems with nonpolarizable dipoles, and is further extended to polarizable ones.

Attempts are also made to relate dielectric and mechanical properties of

polymers.

The time rate of change of a polarization vector in a dielectric isotropic

system is studied in Chapter 3, using extended irreversible thermodynamics. This

is a novel approach not often found in studies of dielectrics. It is shown how

conservation equations in conjunction with the entropy production equationmake it possible to obtain expressions that in principle could describe dielectric

relaxation processes, even in the cases in which the dipoles have one component

parallel and the other perpendicular to the chain contour. In contrast with

classical irreversible thermodynamics, where the equations are parabolic, the

present approach, based on extended irreversible thermodynamics, leads to

hyperbolic equations with finite speed for the propagation of electrical signals.

Taking advantage of the fact that the linear responses in the frequency and

time domains of systems to a step function field are related through Fourier

transforms, experimental devices have been designed that allow determination of

the dielectric behavior of polymers over a frequency/time window of about 12decades. Chapter 4 is focused on the description of these instruments as well as on

the underlying physics. Empirical equations that allow the analysis of the dielec-

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tric results are described in detail. Special attention is devoted to the behavior of

electrets as monitored by thermal stimulated discharge currents.

Owing to the flexibility of molecular chains, including those containingrigid segments in their structure, the square of the dipole moment of a polymer is

the average over the square of the dipole moments associated with the large

number of conformations of the system. Statistical mechanics methods are

described in Chapter 5 that allow the computation of the mean-square dipole

moments of polymers by assuming that the skeletal bonds are restricted to a

limited number of rotational states. The use of this analysis to obtain both mean-

square dipole moments in terms of the chemical structure and the conformational

energies associated with rotational states is emphasized.

Among the conformational properties most sensitive to chemical structure,the electric birefringence expressed in terms of the molar Kerr constant stands

out. Chapter 6 deals with the experimental measurements of the electric

birefringence of polymer solutions and the development of mathematical expres-

sions obtained by statistical mechanical procedures that relate the Kerr constant

with the averages of the polarizability tensors associated with the conformations

of the chains. The procedure for assigning the polarizability tensor of groups of

bonds to each skeletal bond of the chains is illustrated.

Chapter 7 deals with the use of molecular dynamics to compute the

trajectory of the dipole moments of molecules in the conformational space. Thefundamentals of molecular dynamics techniques are given in detail, emphasizing

how the time dipole correlation functions obtained from the trajectories of

monomers and low-molecular-weight polymers can be used to compute their

mean-square dipole moments and their relaxation spectra in the frequency

domain.

Dielectric behavior is an excellent diagnostic property in that it reflects

molecular structure and motions. The wide frequency window available in this

technique makes it possible to obtain isotherms displaying the glass rubber and

the secondary relaxations of polymer melts in the frequency domain. Chapters 8and 9 discuss how the chemical structure of polymers may affect their relaxation

spectra. The relaxation spectra of a few polymers are included and theories

interpreting short- and long-range motions are presented in these two chapters.

A step electric field applied to a polymer solution induces a birefringence in

the system that increases with time as the molecules rotate, reaching a constant

value at equilibrium. Removal of the electric field decreases the birefringence

to zero as the Brownian motions randomize the orientations of the mole-

cules. Chapter 10 deals with the study of the buildup and decay functions and

how these functions are related to the rotational relaxation times of molecular

chains.

Liquid crystals are characterized by the molecules being free to move as in

a liquid, although as they do so they tend to spend a little more time pointing

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along the direction of orientation than along some other direction. Chapter 11

studies the microscopic and macroscopic order parameters of mesophases and

their relation with the permittivity. Theories developed for isotropic systems aremodified to account for the equilibrium and dielectric relaxation behavior of

diverse mesophases. Ferroelectricity in liquid crystals is also discussed.

For certain polymers an intrinsic polarization can be induced by these

effects of stress or temperature. These intrinsic piezoelectric and pyroelectric

materials frequently obtain their anisotropic polarization through some structural

rearrangement involving either crystal packing or dipole alignment of macro-

poles. This is the subject of Chapter 12, where the relationships between the

polarization vector and the stress tensor in piezoelectrics polymers as well as

between temperature and polarization in pyroelectrics are studied. Polymer struc-tures that can develop ferroelectric, pyroelectric, and piezoelectric properties are

discussed.

Polymers containing certain chromophore groups in their structure as well

as ferroelectric materials are being considered promising candidates for future

nonlinear optical (NLO) applications, such as frequency doublers, optical storage

devices, electrooptic uses and modulators. Their advantage over traditional

inorganic materials such as LiNbO3 basically lies in their high laser damage

threshold and their ease of processing and architectural modification. Chapter 13

gives an overview of the physical fundaments of nonlinear optics and second-harmonic generation in polymers, emphasizing the physics underlying the rela-

tions between second-order susceptibility and hyperpolarizability, poling decay,

etc. Attention is paid to the guidelines that allow the design of polymeric systems

containing chromophore groups with good NLO properties.

The synthesis of conductors or semiconductors that retain the desirable

polymeric attributes of moldability, flexibility, and toughness is a subject of great

importance from a basic and applied point of view. Chapter 14 describes double

bond conjugated polymers that conveniently doped could produce good

electronic conduction. Semiempirical quantum mechanics methods useful forthe computation of the energy gaps between the valence and conduction bands

are discussed. The conduction mechanisms in the doped conducting polymers

and the nature of the conducting species in the doped polymers are studied.

Attention is also paid to the use of these materials in electroluminiscence,

batteries, electromagnetic interference shielding, anticorrosion, etc.

This book brings together the coverage of different electrical phenomena in

polymers and of how both chemical and the supermolecular structures may affect

them. The book is not intended to be an overall review of electric phenomena but

rather a description of the fundamentals of these phenomena in relation to the

structure of polymers. Some chapters, especially the basic ones, include problem

sets that we hope will facilitate the understanding of the subjects discussed in the

book, especially for readers who are not familiar with the dielectric behavior of

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polymeric materials. Some of these problems deal with important aspects of the

theory not fully developed in the main text. This book can also be used as a

textbook in undergraduate and graduate courses of materials science.

Evaristo Riande

Ricardo Daz-Calleja


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Table of Contents

Preface

Part I. Physical Fundaments of Dielectrics

Chapter 1. Static Dipoles

1.1. Dipoles

1.2. Electric potentials arising from an isolated dipole

1.3. Point dipole

1.4. Field of an isolated point dipole

1.5. Force exerted on a dipole by an external electric field

1.6. Dipole dipole interaction

1.7. Torques on dipoles

1.8. Dipole moment and dielectric permittivity. Molecular versus

macroscopic picture

1.9. Local field. The Debye static theory of dielectric permittivity

1.10. Drawback of the Lorentz local field

1.11. Dipole moment of a dielectric sphere in a dielectric medium

1.12. Actual dipole moments, definition and status

1.13. Directing field and the Onsager equation

1.14. Statistical theories for static dielectric permittivity.

Kirkwoods theory

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1.15. Frohlichs statistical theory

1.16. Distortion polarization in the Kirkwood and Frohlich theories

Appendix A Properties of the Legendre polynomialsAppendix B Frohlich alternative calculations for the polarization

of a dielectric sphere in an infinite medium

Appendix C The polarization of an ellipsoid

Appendix D Important formulae in SI units

Problems

References

Chapter 2. Quasi-static Dipoles

2.1. Brownian motion

2.2. Brief account of Einsteins theory of Brownian motion

2.3. Langevin treatment of Brownian motion

2.4. Correlation functions

2.5. Mean-square displacement of a Brownian particle

2.6. Fluctuation dissipation theorem

2.7. Smoluchowski equation

2.8. Rotational Brownian motion

2.9. Debye theory of relaxation processes

2.10. Debye equations for the dielectric permittivity

2.11. Macroscopic theory of the dielectric dispersion

2.12. Dielectric behavior in time-dependent electric fields

2.13. Dissipated energy in polarization

2.14. Dispersion relations

2.15. Energy dissipation and the Debye plateau

2.16. Inertial effects

2.17. Langevin equation for the dipole vector

2.18. Diffusive theory of Debye and the Onsager model

2.19. Relationship between macroscopic dielectric and mechanical

properties

2.20. Statistical mechanics and linear response

2.21. Relationship between the frequency-dependent permittivity

and the autocorrelation function for dipole moments

2.22. Extension to polarizable dipoles

2.23. Macroscopic and microscopic correlation functions

2.24. Complex polarizability

2.25. Dispersion relations corresponding to the polarizability.

A new version of the fluctuationdissipation theorem

2.26. Fluctuations in a spherical shell

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2.27. Dielectric friction

2.28. Resonance absorption

2.29. Memory functions2.30. First-order memory function and macroscopic relaxation time

2.31. Mode coupling theories

Problems

References

Chapter 3. Thermodynamics of Dielectric Relaxations

in Complex Systems

3.1. Thermodynamics of irreversible processes

3.2. Dielectric relaxation in the framework of LIT

3.3. Maxwell equations

3.4. Conservation equations

3.4.1. Conservation of mass

3.4.2. Conservation of charge

3.4.3. Conservation of linear momentum

3.4.4. Conservation of energy

3.4.5. Internal energy equation3.5. Entropy equation

3.6. Relaxation equation

3.7. Correlation and memory functions

3.8. Dielectric relaxations in polar fluids

3.8.1. Introduction

3.8.2. Balance equations

3.8.3. Entropy equation

3.9. Dielectric susceptibilities and permittivities

3.10. Generalization and special cases3.11. Memory function

3.12. Normal mode absorption

Appendix

Problems

References

Chapter 4. Experimental Techniques

4.1. Measurement systems in the time domain

4.2. Measurement systems in the frequency domain

4.3. Immittance analysis

4.3.1. Basic immittance functions

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4.3.2. Series and parallel RC networks

4.3.3. Mixed circuit. Debye equations

4.4. Empirical models to represent dielectric data4.4.1. Retardation time spectra

4.4.2. Cole Cole equation

4.4.3. Fuoss Kirkwood equation

4.4.4. Davidson Cole equation

4.4.5. Havriliak Negami equation

4.4.6. Jonscher model

4.4.7. Hill model

4.4.8. KWW model

4.4.9. Dissado Hill model4.4.10. Friedrich model

4.4.11. Model of Metzler, Schick, Kilian, and Nonnenmacher

4.4.12. Biparabolic model

4.5. Thermostimulated currents

4.5.1. Electrets

4.5.2. Thermostimulated depolarization and polarization

4.5.3. Microscopic mechanisms and applications of TSD

currents

4.5.4. Basic equations for dipolar depolarization4.5.5. Isothermal measurements

4.5.6. Thermal windowing

4.5.7. Thermostimulated depolarization currents versus

conventional dielectric measurements

Appendix A Edge corrections

Appendix B Single-surface interdigital electrode

Problems

References

Part II. Structure Dependence of the Equilibrium and Dynamic

Dielectric Properties of Polymers

Chapter 5. Mean-Square Dipole Moments of Molecular Chains

5.1. Introduction

5.2. Dipole moments of gases

5.3. Dipole moments of liquids and polymers

5.4. Effect of the electric field on the mean-square dipole moment

5.5. Excluded volume effects

5.6. Dipole moments for fixed conformations

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5.7. Average values ofm2

5.8. Rotational states and conformational energies

5.9. Dipole autocorrelation coefficient of polymers5.10. Determination of conformational energies

References

Chapter 6. Electric Birefringence of Polymers under Static Fields

6.1. Introduction

6.2. Birefringence: Basic principles

6.3. Electric birefringence6.4. Induced dipole moments and polarizability

6.5. Orientation function of rigid rods

6.6. Evaluation ofmK for polymer chains

6.7. Realistic model for the evaluation of mK in flexible polymers

6.8. Valence optical scheme

6.9. Computation of mK by the RIS model

Problems

References

Chapter 7. Molecular Dynamics Simulations of Equilibrium and

Dynamic Dielectric Properties

7.1. Introduction

7.2. Basic principles of molecular dynamics

7.2.1. Force fields

7.2.2. Integration algorithms and trajectories

7.2.3. Computation time savings7.3. Trajectories of molecules in phase space and computing time

7.4. Determination of the time dipole correlation coefficient

References

Chapter 8. Dielectric Relaxation Processes at Temperatures

Above Tg Molecular Chains Dynamics

8.1. Introduction

8.2. Phenomenological dielectric response in the time domain

8.3. Dielectric response in the frequency domain

8.4. Dielectric relaxation modulus in the time and frequency

domains

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8.5. Kronig-Kramers relationships

8.6. Analytical expressions for the dielectric permittivity and

dielectric relaxation modulus in the time and frequencydomains

8.7. Local and cooperative dynamics: Basic concepts

8.8. Responses of glass formers above Tg to perturbation fields in

a wide interval of frequencies

8.9. Broadband dielectric spectroscopy of supercooled polymers

8.10. Temperature dependence of the stretch exponent for the

a-relaxation

8.11. Temperature dependence of secondary relaxations

8.12. Temperature dependence of the a-relaxation8.13. Dielectric strength and polarity

8.14. Segmental motions

8.15. Long-time relaxation dynamics

8.16. Time dipole correlation function for polymers of type A

8.17. Normal relaxation time for polymers having type A and

type AB dipoles

8.18. Molecular chains dynamics

8.18.1 Rouse model

8.18.2. Zimm model8.19. Normal mode relaxation time for melts and concentrated

solutions of polymers having type A and type AB dipoles

8.20. Scaling laws for the dielectric normal mode of semi-dilute

solutions of polymers having either type A or type AB dipoles

8.21. Relation between molecular dimensions and relaxation

strength for type A polymers

References

Chapter 9. Relaxations in the Glassy State. Short-Range Dynamics

9.1. Introduction

9.2. Molecular models associated with a single relaxation time

9.3. Dynamics of secondary dielectric relaxation in two-site

models

9.4. Coalescent ab-process

9.5. Relaxation ofN segments in a chain: Dynamic rotational

isomeric state model (DRIS)

9.6. Probability of rotational states at equilibrium

9.7. Conformational transition rates

9.8. Independent conformational transitions

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9.9. Pairwise dependent conformational transitions

9.10. Time dipole autocorrelation coefficient

9.11. Relaxation times9.12. Motion of a single bond

9.13. Phenomenological classification of secondary relaxation

processes

9.13.1. Local main chain motions

9.13.2. Motions of side groups about their link to the main

backbone

9.13.3. Motions within side groups

9.13.4. Motions due to the presence of small molecules in

the polymer matrix9.13.5. Secondary relaxations in semicrystalline polymers

9.13.6. Dielectric relaxations in liquid crystalline polymers

(LCPs)

Problems

References

Chapter 10. Electric Birefringence Dynamics

10.1. Introduction

10.2. Orientation function

10.3. Decay function

10.4. Buildup orientation function

10.5. Pulsed fields

10.6. Dispersion of the birefringence in sine wave fields

Problems

References

Part III. Special Polymers

Chapter 11. Dielectric Properties of Liquid Crystals

11.1. Introduction: Liquid crystal generalities

11.2. Tensorial dielectric properties of anisotropic materials

11.3. Macroscopic order parameter

11.4. Microscopic order parameter

11.5. Dielectric susceptibilities

11.6. High-frequency response

11.7. Static dielectric permittivities

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11.8. Smectic phases

11.9. Dynamic dielectric permittivity. Internal field factors

11.10. Dielectric relaxations in nematic uniaxial mesophases11.11. Experimental results

Appendix A

Appendix B

Appendix C

Problems

References

Chapter 12. Piezoelectric and Pyroelectric Materials

12.1. Introduction

12.2. Basic concepts

12.3. Thermodynamics

12.4. Piezoelectricity

12.5. Pyroelectricity

12.6. Piezoelectric and pyroelectric polymers

12.7. Piezoelectricity effect and symmetry

12.8. Piezoelectric and pyroelectric mechanisms12.9. Piezoelectricity and pyroelectricity in polar polymers

12.10. Uniaxially oriented, optically active polymers

12.11. Ferroelectric liquid crystalline polymers

12.12. Measurements of piezoelectric constants

12.13. Relation between the remnant polarization and the

piezoelectric constants in ferroelectric polymers

12.14. Ferroelectric composites

References

Chapter 13. Nonlinear Optical Polymers

13.1. Introduction

13.2. Basic principles of harmonic generation in crystals

13.3. Second harmonic generation and coherence length

13.4. Nonlinear polarization and frequency mixing

13.5. Nonlinear polarization in polymers

13.6. Poling process

13.7. Relation between hyperpolarizability and second-order

susceptibility in uniaxially poled oriented polymers

13.8. Poling decay

13.9. Determination ofx2ijksusceptibilities

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13.10. Measurement of the electrooptic effect

13.11. Guidelines for designing NLO polymer systems

13.12. Polymer systems with NLO propertiesReferences

Chapter 14. Conducting Polymers

14.1. Introduction

14.2. Chemical structure and conducting character

14.3. Routes of synthesis of conjugated polymers

14.3.1. Electropolymerization14.4. Energy gaps in conducting polymers

14.5. Doping processes

14.6. Charge transport

14.7. Metallic conductivity

14.8. Microwave dielectric permittivity

14.9. Optical dielectric permittivity and conductivity

14.10. Applications of semiconductor polymers

14.11. Conducting polymers

14.12. Polymers for rechargeable batteries14.13. Other applications

Problems

References
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